Office 365 - Server-side vs Client-only Rules
Rules are conditions/actions set upon incoming and outgoing messages. You can divide rules into two types - server-side and client-only. Server-side rules are handled entirely by the Exchange Server (Office 365), independent of the state of the Outlook client. Client-only rules do not execute until the user who created the rule logs in to the Outlook client with the same profile used to create the rule.
If you are experiencing a problem with one of your rules, you can verify if the rule is a server-side or client-side by seeing if the rule exists by seeing if you can manage it via Outlook on the web. If you see it within Outlook on the web, then it is a server side rule.